Optimal. Leaf size=63 \[ \frac {\left (a+b x+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)} \]
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Rubi [A]
time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {708, 272, 67}
\begin {gather*} \frac {\left (a+b x+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d^3 (p+1) \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 708
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^p}{x^3} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x}{4 c d^2}\right )^p}{x^2} \, dx,x,(b d+2 c d x)^2\right )}{4 c d}\\ &=\frac {(a+x (b+c x))^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 64, normalized size = 1.02 \begin {gather*} \frac {(a+x (b+c x))^{1+p} \, _2F_1\left (2,1+p;2+p;\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+b x +a \right )^{p}}{\left (2 c d x +b d \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\left (a + b x + c x^{2}\right )^{p}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (b\,d+2\,c\,d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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